US 

:opy 1 



A STUDY OF THE PLOW BOTTOM AND ITS ACTION 
UPON THE FURROW SLICE 



A THESIS 

Presented to the Faculty of the Graduate School 

OF Cornell University for the degree of 

DOCTOR OF PHILOSOPHY 



E. A. WHITE 



Reprinted from Journal of Agricultural Research, Vol. XII, No. 4, January 28, 1918 



A STUDY OF THE PLOW BOTTOM AND ITS ACTION 
UPON THE FURROW SLICE 



A THESIS 

Presented to the Fac(tlty of the Graduate School 

OF Cornell University for the degree of 

DOCTOR OF PHILOSOPHY 



BY 

E. a: WHITE 



Reprinted from Journal of Agricultural Research, Vol. XII, No. 4, January 28, 1918 



,• u 



{J. k-t 



w* 



al 



jQ^-ioli 



A STUDY OF THE PLOW BOTTOM AND ITS ACTION 
UPON THE FURROW SLICE' 

By E. A. White, - 

Assistant Professor of Farm Mechanics 

College of Agriculture of the University of Illinois 

INTRODUCTION 

The most ancient records show that from a very remote period man 
has used the plow, in one form or another, to assist him in stimulating 
the earth to bring forth a more bountiful harvest. As has been the 
case in many other Hnes of endeavor, theory has trailed far behind 
observation and experience in developing this implement. In fact, as 
far as can be ascertained, it was not until the last half of the eighteenth 
century that any serious attempt was made to develop a plow bottom 
from a theoretical standpoint, and even then the productions of Jeffer- 
son, Lambruschini, Small, Rham, and others can not be considered as 
thoroughly grounded upon well-developed theories; rather their works 
should be looked upon as hypotheses (fig. i). Experience in the field 
generally proved that the machines designed by these men were not all 
that could be desired — for example, it is reported ^ that when Lam- 
bruschini's helicoidal moldboard was taken into the field for trial the 
driver of the draft animals immediately observed that the force required 
to move this plow was too great for the results obtained. To be sure, 
geometrically exact moldboards furnished the basis in many instances 
for more perfect developments, but the results obtained by empirical 
plow designers who worked in the field were so far superior to the results 
obtained by the men who worked in the laboratory that the theorists 
were soon completely outstripped and even held up to ridicule by the 
men who developed their machines in the hard school of experience, 
until at the present time we find special types of plow bottoms designed 

» Approved for publication in the Journal of Agricultural Research by the Director, Cornell University 
Agricultural Experiment Station. 

2 The experimental work for this paper was done under the direction of Prof. H. W. Riley, of the De- 
partment of Rural Engineering, Cornell University, and the mathematical developments were prepared 
under the supervision of Prof. F. R. Sharpe, of the Department of Mathematics. In addition to the above, 
gratelul acknowledgments are given to the following: To Profs. James McMahoi. and Virgil Snyder, of 
the Department of Mathematics, for their most timely and helpful suggestions; to Mr. J. E. Reyna, 
Instructor in Drawing, College of Agriculture, Cornell University; and to Mr. L. S. Baldwin, Instructor 
in General Engineering Drawing, University of Illinois, for making the drawings. 

'Lambruschini, R. d'un nuovo orecchio oa coltri. In Gior. Agr. Toscano, v. 6. p. 37-80. 1832. 



Journal of Agricultural Research. Vol. XII, No. 4 

Washington, D. C. Jan- 28, 191S 

It Key No. N. Y. (Cornell)— 3 

(149) 



I50 



Journal of Agricultural Research 



Vol. XII, No. 4 



to meet certain field conditions; but no well-developed theory is avail- 
able to serve as a guide in this work. 

This paper is an attempt to begin a fundamental analysis of the plow 
bottom and its work, in the hope that some light may be thrown upon the 
theory of this humble but perplexing machine, and other attempts stimu- 
lated to delve further into the secrets which are still to be revealed 
regarding the theory of this important implement. Empirical methods 
have given the world plow bottoms which work well. It is still to be 
hoped that scientific investigation can refine and further perfect, supple- 
ment as it were, the productions of experience. 

The work undertaken by the writer can be naturally divided into three 
parts: (i) A study of the forms of plow bottoms; (2) an attempt to 



Date 


Name 


Generatrix 


Directrixes 


Equation ofSurhct 




Small 


5tn:iight line 


Straightline & Catenary 






Stephens 


Stnslghf line 


Straight line and 
arc of Circle 


^^tanffiz); 


1768 


JefFerson 


Straight Une 


Straight lines 


l^//=^- 


teia 


Day/5 


/Ire of Circle 


/Ires ofCin:le 




I63Z 


Lambruschini 


5tra/gh/ Line 


Straight line and 
tieliK 


^ = f-anfaz) 


1639 


W/theroiy 
andP/erce 


Arc ofCyc/o/c/ 


Arcs ofCycioiaf 




1640 


Rham 


Straight line 


Straight l/nes 


a' b^-^^^ 


1340 


Rham 


Straight- line 


Curves 




/852 


Knox 


Straight line 


Arcs of Circles 


fFu/ed sunifce of 
el&hrh order 


/654 


Gibbs 


Straight line 


Arcs of Circles 


f.^f.-/=o 


1663 


f\/1eaof 


Straightline 


Arcs of Circles 


a^^b'' C^ '^ 


/667 


tiolbrook 


Straightline 


Srraight Ijhes 


a'^3' c'-' 


1834 


Jacobs 


/f port/or? ffTom each of £ S or/aces; each surface 
haWng 2 sets ofstralghf fine generators. 



Fig. I. — Diagram giving the generatrices, directrices, and equations of surfaces of historical plow bottoms . 

analyze the motion of the soil particles as they pass over the surface, 
and (3) a mathematical analysis of the surfaces of the most important 
historical plow bottoms which were designed to be geometrically exact. 
It was, and still is, hoped that a knowledge of just what the plow bottom 
is and how it performs its work will be of material assistance in developing 
a theory which will furnish a very definite basis for the proper design of 
this fundamental implement of tillage. 

FORMS OF THE PLOW BOTTOM 

A study of modern American-manufactured plow bottoms reveals 
the fact that a large number of these are so constructed that their surfaces 
contain sets of straight lines, each set consisting of an infinite number of 
straight lines, so related that an equation or equations satisfied by the 
coordinates of points on the surface can be found. 



jaa. »8, 1918 Study of Ploiv Bottoms I C I 

Plate 6, A, represents a bottom with two sets of straight lines. The 
few lines shown in the illustration indicate that through every point of 
the surface two straight lines can be drawn which lie wholly on the surface 
until they pass off the edges of the bottom. These straight lines furnish 
the basis for the proof that such a surface is a portion of an hyperboloid 
of one sheet (for the form of this surface see fig. 3 to 7) whose equation 
can be developed and studied with mathematical exactness. The 
method of developing this equation will be given later, but at present we 
are mainly interested in the fact that there is a classs of plow bottoms on 
whose surfaces lie sets of straight lines, and, further, that one equation can 
be developed which will approximately represent the working surface of 
such a bottom. 

Further study shows that the surfaces of other plow bottoms contain 
sets of straight lines, but that one equation will not completely describe 
such a surface. In Plate 6, B, a bottom is shown whose surface is com- 
posed of a portion of each of two surfaces. Plate 6, C, shows a similar 
bottom, but in this case the two surfaces merge into each other farther 
back upon the moldboard. 

In Plate 6, D, a class of bottoms is represented whose entire surfaces 
do not contain an infinite set of straight lines. It is true that the share 
and back end of the moldboard exhibit the same characteristics that the 
first two classes have shown, but the lines do not continue to the fore part 
of the moldboard. 

Plate 7, A, shows a plow bottom with a convex surface which has two 
sets of straight lines. 

The American-manufactured plow bottoms studied can thus be 
divided into three general classes: (i) A portion of one quadric surface; 
(2) a portion of each of two quadric surfaces, and (3) nonquadric sur- 
faces. Nearly all forged bottoms belong to classes i and 2 with the 
majority falling into class 2, while most of the cast bottoms belong to 
class 3. It should be noted, however, that some recently designed cast 
bottoms depart from the general characteristics of class 3 and show 
clearly the two quadric surfaces of class 2. The lines running in the 
general direction, front to rear, marked "/," (PI. 6, A) will be called 
longitudinal lines, and those running in the general direction, top to 
bottom, marked "t" (PI. 6, A) will be called transverse lines. 

For the purpose of studying the forms of the various surfaces under 
consideration, a machine, illustrated in Plate 7, B, was designed and built 
for measuring the space coordinates of any desired point.* By means 
of slots and a system of pulleys attached to the drafting board the cross- 
bar can be kept horizontal and be moved both laterally and vertically, 
while the drafting board is attached to a frame which can be moved 

' Similar machines are described in the following publications: Ck)Ui,D, J. S., et al. report on Triais , 
OF PLOWS. In Trans. N. Y. State Agr. Soc, v. 27, pt. i. 1867, p. 426. 1868. 
Giordano, Federigo. le ricerche sperimentai,i di meccanica agraria. p. ho. Milano, 1906. 



152 



Journal of Agricultural Research 



Vol. XII. No, 4 



backward and forward upon guides so marked that the board in all posi- 
tions will be squarely across the guides. When a plow bottom is properly- 
placed upon the platform the x, y, and z coordinates of any point upon 

the surface can thus be recorded upon coor- 
dinate paper fastened " upon the drafting 
board. 



/ 



/ 






Q i^zVz^a 



X 



y 






DEVELOPMENT OF THE EQUATION 



j(,y,z 



.^xy2' 



From a mathematical standpoint the sur- 
face shown in Plate 6, A, presents the 
problem of finding the equation of a surface, 
given two sets of straight-line generators. 
This can be done if the equations of any 
three lines in the same set are known. 
Select three lines ah, cd, and ef (fig. 2). 
Let Xj, y^, z\, and Xn, ^2. ^2 be the coordinates 
of two points upon line ab; 3C3, y^, z^, and x^,y^, z^ of two points upon line 
cd; and x^, y^, z^, and x^, y^, z^ of two points upon line e/. 
The equations of the lines ab, cd, and ef, are 



Fig. 2. 



and 



x- 


-■^1 


y- 


-.Vi 


z - 


-21 


Xn- 


-^1 


}'2 ' 


->'i 


Zo- 


-21 


X.- 


-X3 


y- 


->'3 


z - 


"2^3 


^4- 


-^.3 


y^- 


->'3 


^■t~ 


-23 


X- 


-^5 


y- 


->'5 


z - 


-^6 



•>'5 



(I) 



(2) 



(3) 



From (2) the following equation for a plane perpendicular to the XY- 
plane and containing the line cd is obtained : 



n=(x - %) ()/, - y^) - (y - y^) {x, - x^) = O. 



(4) 



Similarly from (2) the equation of a plane perpendicular to the YZ- 
plane and containing the line cd is 



«5=0'->'3) (24-2.0- (2-23) iyi-yz) = o. 



(5) 



From (3), the equation of a plane perpendicular to the Xl^-plane and 
containing the line ef is 



2*6= {x - ^5) O'o - yd -iy-yd{\-xd=o. 



(6) 



Similarly, from (3) the equation of a plane perpendicular to the KZ-plane 
and containing the line ef is 



^7= (y - yd (26 - 25) - (2 - 25) (je - yd = o. 



(7) 



Jan. 28, i9i8 Study of Plow Bottoms 153 

Consider 

u^ = Au^. (8) 

where A is a constant. This is the equation of a plane which contains 
the intersection of planes (4) and (5); hence it contains the line cd. 
Similarly 

Uq = Bu^ (9) 

where B is a constant, is the equation of a plane which contains the line ef. 
If A and B have such values that the point x', y' , z' is on (8), (9), and 
(i), the line of intersection of (8) and (9) meets (i) and is a generator 
(see fig. 2). Hence, 

{y'-yz){^-z^)-{^'-z,){y,-y,)' ^'°'' 



and 



^ {x'- %^ { y^ -y^-iy'- ys) JH - ^5) . ,. 

iy'-y.W,-^.)-{z'-z,){y,-y,) ' ^"^ 

^l^^=t^::y^=t^=K; (12) 

^1-^1 y^-yx ^2-21 ' 

where A' is a constant. 
From equations (12) 

%' = K{%^-%^)^x^ (13) 

y'=K{y2-yi)+yi (14) 

z' = K(z2-z,)+Zi (15) 
From equations (10), (13), (14), and (15) 

^^ i[K(x2-Xi)+Xi-X3\(y,-ys))-i[K{y.,-yj)+yi-y3](x,-X3)) _ . ^. 

{[K{y2-yi)+yi-y3]{Zi-Z3))-{[K{22-Zi)+Zi-z3]{yi-y3)) ' ^ ^ 

and from equation (8) 

^^^ (x-X3)iy,-y3)-(y-y3)iXi-X3) , . 

From equations (11), (13), (14), and (15) 

{\K{ x.-x;)-\-x^-x^{yf,-y^y)-(\K(y^-y^)-^y^-y^{x^-x^) 

and from equation (9) 

^_ (.r - xQlJ'e - Vs) - (y - >'5)(^6 - ^5) / N 

(j'->'5)(2b-25)-(s-25)()'e-y5) ^ ^-^ 

Eliminating A,B, and /v from (16), (17), (18), and (19), we have the 
equation of a surface through the lines ah, cd, and ef. The equations are 
left in this form because numerical substitutions are more easily made 



154 Journal of Agricultural Research voi. xii. no. 4 

at this point than would be the case if the indica<:ed operations were first 
performed with the symbols.^ The general form of the equation resulting 
from the previous operations is 

ax^ + hy^ + cz' + 2fyz + 2gxz + 2 hxy -\-2lx-\- 2my -\-2nz-\-d=0. (20) 

To reduce equation (20) to its simplest form the axes must be trans- 
lated and rotated. 

TRANSLATION OF AXES^ 

The origin of equation (20) is translated to the center by putting 

x=x'+Xa, y=y'+yo, z+z'+z^; (21) 

he values of Xg, Vq, and Zo being obtained from the following: 

aXo-^hyo + gZo+l = (22) 

hxo + hyo+fzo + m = (23) 

gXo+fyo+cza + n = 0. (24) 

These substitutions give, after dropping the accents from x', y', and 2', 
an equation of the following form : 

ax"^ + by^ + c^ + 2/;'2 + 2qxz + 2 hxy -\-G = 0\ (25) 

where G=-lx^^myo-VnZo + d. (25a) 

ROTATION OF axes' 

Equation (25) can be further reduced by a rotation of the axes. This 
is accomplished by means of a cubic equation 

/f.^-(a+6+c)A;2 + (a6+ac + tc-f-(r-/r)^-^ = 0; (26) 



where D- 



a 


h 


9 


h 


b 


/ 


9 


f 


c 



(26a) 



I^et the roots of (26) be k^, k^, and k^. The desired equation, after trans- 
lating and rotating the axes is 

A 
k,x^ + k.y + ^'322 4- ,-, -r- = O; * (27) 

' A numerical problem is developed by this method upon pages 156 to 160. 

* Snyder, Virgil, and Sisam, C. H. analytic geometry op space,, p. 77. New York, 1914. 

'Idem, p. 79. 

*Idem, p. 86. 



Jan. 28, 1918 



Stvdy of Plow Bottoms 



155 




Skeleton. Hyperboloid of One 5haet 
Fig. 3. 




Section z=o. Fig 3. 
Fig. 4. 




Section y = 0, Fig. 3. 
Fig. s. 




o y 




Section x^>=o. Fig 3. 

Fig. 6. 



Hyperboloid of One Sheet, showing Lines 
upon the Surface 



Fig. 7. 



156 Journal of Agricultural Research voi. xii. no. 4 

where A = DG. (27a) 

The direction cosines X, ju, v, of the angles which the new X-axis makes 
with the original axes are obtained from the following : 

(a—ki)\ + hfjL+gv = (28) 

h\-{-{h-k^)n+jv = (29) 

pX+//i + (c-fe,)i; = C> (30) 

^\+lx'' + v''=l. (31) 

Similarly, the direction cosines of the angles which the V- and Z-axe 
make, after rotation, with the original axes are found by substituting 
feg and ^3, respectively, for k^ in equations (28), (29), (30), and (31). 

When equation (27) was developed from the surface of a plow bottom 
having two sets of straight-line generators, it had the following general 
form: 

X^ /,,2 2^ 

This is the equation of an hyperboloid of one sheet, a vase-shaped figure, 
the skeleton of a section of which is shown in figure 3. When z = 

equation (32) becomes -^ + p=i, and the cross section through the 

plane z = (fig. 4) is an ellipse. When y = 0, the equation becomes 

x^ z- 

-2 — 2~ i> ^"^^ the section through the plane y = (fig. 5) is a hyperbola. 

\~ z^ 
Similarly, when x = 0, f^. — -1= i (fig- 6). Figure 7 indicates the two sets 

of straight-line generators which lie on the surface of an hyperboloid of 
one sheet.^ 

APPLICATION OF THE DEVELOPMENT TO A PROBLEM 

In order to^develop the equation which will describe the surface of a 
plow bottom, it is necessary to obtain the data called for in equations 
(16), (17), (18), and (19). This application of the development will be 
carried through for the bottom represented in Plate 6, A, which bottom 
was placed upon the machine shown in Plate 7, B, so that the origin of 

'The constants a, b, and c ol this equation do not necessarily have the same numerical values as in 
previous equations. 

' The method for obtaining the equations of any line on the surface is given in Snyder, Virgil, and 
SiSAM, C. H. Op. cit., p. 93. 



Jan. 28, 1918 



Study of Plow Bottoms 



157 



coordinates came at O, figure 8. The plane y = contains the points O, 
m, and n; and the plane x = contains the points O and m and is per- 
pendicular to the plane y = 0. The plane 2 = is perpendicular to both 
the planes y = and x-=0. The axes are considered to be positive in the 
directions indicated by the arrowheads (fig. 8). Three transverse lines, 
ab, cd, and ef (fig. 8), were selected and the following data obtained: 




Fig. 8. 



Table J.— Values {in inches) developed for the surface of the plow bottom shown in 

Plate 6, A 



Xi= 2.84 

>'i= 5-7 
2i=i6.o 


Xo= 7-42 

>'2= 3-78 

22=19.0 


^3= 4-42 

y,= 8.74 
23=20.0 


^4= 8.54 
>'4= 6.43 

24=23.0 


^5= 9-7 

J'5= 10.88 

25=26.0 


X6=I2.58 

J6= 7-65 
2^ = 28.0 



When the above values are substituted in the equations already 
developed, 
From (16) 

-2.287^ + 13-83 
K-15.7 • 



^ = 



(33) 



158 



Journal of Agricultural Research 



Vol. XII, No. 4 



From (17) 

From (33) and (34) 

From (18) 

From (19) 

From (36) and {2>7) 
K- 



A = 



■X— 1.783/ + 20 



i.299:i/+2-3i-35 

-i5. 73g+io.o4>/- 13.832+ 1 1.95 
X- 1. 177>'- 2.282 + 51.45 

-1.584/C + 6.335 



B 



B- 



K-7.29 
— X— .892>'+i9.4 



. 6197+2- 32.74 * 

7. 29a; + 2. 587-6.3352 + 65.85 
.1;+. 0887— 1.542 + 32.46 



(34) 



(35) 



(36) 



(37) 



(38) 



By eliminating K from equations (35) and (38) the following equation 
for the surface of the plow bottom is obtained: 



3.gx-+y^- + T,.4Sz"-7.ssyz-7-^^xz + 6.79xy 

+87. irc+ 1 20.757- 75.052+ 227. 25= a 



(39) 



Table II is compiled for purposes of checking the values computed from 
equation (39) with those obtained by measuring. 

Table II. — Values (in inches) for the surface of the plow bottom shown in Plate 6, A, 

obtained by measurement 









X computed 




z 


y 


X 


from Differ 
equation (39) 


ence. 


10 


2 


2.9 


2. 27 


63 


15 


6 


1-53 


1.56 - 


03 


15 


4 


3-58 


3-77 


19 


15 


2 


6.9 


6.32 


,=;8 


20 


10 


3-72 


3-8 


08 


20 


8 


4-73 


4. 76 - 


03 


20 


4 


7.83 


7-94 


13 


25 


12 


8. 22 


8.12 


I 


25 


9 


9.07 


9.2 - 


13 


25 


6 


10.43 


10. 46 — 


03 


30 


10 


14 


13.86 


14 


32 


9 


16.5 


16. I 


4 



Jan. 28, 1918 



Study of Plow Bottoms 



159 



To find the geometric center, substitute the coefficients from equation 
(39) into equations (22), (23), and (24). Solving, we find 

Xo= — 1.405 inches. 
j/o= 6.52 inches. 
2^0= 16.4 inches. 

This translation of axes is shown in figure 9. From equation (25a) 
G=— 57.3. From (25) the equation of the surface referred to parallel 
axes through the center is 

3.9a;2 + ;^'2 4- 3452^- 7-53>'2- 7-28x2+ 6.79x^-57.3 = 0. (40) 

k 

\y 



<s 




/ / .r: A 




\ 






■i 





Fig. 9. 

To find the equation of the surface referred to the principal axes through 
the center, substitute the coefficients from (39) into (26), and we have 

fe^— 8.35^—20.17^+2.45 = 0. (41) 

On solving by Horner's method 

^1= 10.27 
^2= 0.128 
/j3= -2.05 

Substituting the values just found for h^, fe,' ^3' -^> ^"^ ^ ^" equation 
(27), we find 

io.27x^+ .i28)r — 2.052:^ = 57.3 

or 

9 « *> 

The direction cosines of the angles which the axes make after rotation 
with the original axes are obtained by making the proper substitutions in 
equations (28), (29), (30), and (31). 



i6o 



Journal of Agricultural Research 



Vol. XII. No. 4 



For the K-axis 



For the Z-axis 



For the X-axis 

7= T0.6136 
/!= T0.48 
u= ±0.627. 

7= ±0.7515 
/x==Fo.i437 
u= ±0.6445. 

7= =Fo.i4i5 
/x= ±0.828 

y== ±0.5425. 

Figure 10 shows the axes after translation and rotation and the por- 
tion of the hyperboloid of one sheet which is a close approximation 

to the surface of this plow 
bottom. 

SURFACES ONE PORTION FROM 
EACH OF TWO QUADRIC 
SURFACES 




Fig. 10. 



were obtained from the share 
board. 



By the use of the method 
which has just been em- 
ployed to develop the equation 
of the surface of the plow 
bottom shown in Plate 6, 
A, two equations can be 
developed which will approx- 
imately represent the surface 
of the bottom shown in 
Plate 6, B. By taking the 
origin as at O, figure 8, the 
data of Tables III and IV 
and the front portion of the mold^ 



Table III- — Values {in inches) developed for the surface of the share and front portion 
of the moldboard of the plow bottom shown in Plate 6, B 



x,= 3.92 
>■!= -8 

2i= 8.0 


.X2= 7-4 
>'2= -75 

2._>=I2.0 


^3= 1-73 

>'3= 2.67 

23=12.0 


^4= 6.78 

3'4= 1-75 
24=16.0 


%= 2.36 
25 = 15-0 


^6= 587 

)'6= 2.7 

26=17.0 



0.25*' + 2. 34>^ +0.462'— 3.25^2— o.773(;2 + 2.66x>/ 

+ 6.88% + 32.3>'- 5.812- 4.4 = 



(43) 



Jan. 28, 1918 



Study of Plow Bottoms 



161 



Table IV. — Values {in inches)for the surface of the share and front part of the moldboard 
of the plow bottom shown in Plate 6, B, obtained by fneasuring 



2 


y 


X 


X computed 
from equa- 
tion (43) 


Difference 


10 


I 


4-75 


4-75 


0. 00 


10 


2 

I 


1-75 
8-37 


I- 54 
9. 00 


. 21 
- -63 


15 


2 


5-47 


5-64 


- .17 


^5 


3 


3-77 


3.82 


- -05 


'^S 


4 


I. I 


1-3 


. 2 



From the remaining surface of the moldboard the following data of 
Tables V and VI were obtained : 



Table V. — Values {in inches) of rest of surface of moldboard shown in Plated, B 



x,= 8.67 


X2= 4-96 


•1^:3=1 1 -73 


3'i= 4-95 


.r,= 8.64 


>3= 4.81 


2, = 24.0 


£2=22.0 


23 = 29.0 


x^= 9.08 


.r5= 13.62 


.r6=I2.24 


^4= 9° 


>'5= 6.23 


.r6=ii.89 


24 = 27.0 


25=33-0 


2<i=3i-o 



1.07%^— I .ojy^ + z^ — T,.ggyz — 1.5x2+ 16. 37X}' 

+ 60.55^^+125.37-48.52+ 109.5 = (44) 

Table Vl.— Values (in inches) for the rest of the moldboard surface shown in Plate 6, B, 

obtained by measurement 









I computed 




z 


y 


I 


from 
equation (44) 


Difference. 


20 


2 


8.85 


8.68 


0. 17 


20 


4 


6.67 


6.78 


— . II 


20 


6 


4.9 


4-95 


- -05 


20 


8 


3-4 


3-5 


— . I 


25 


3 


10. 6 


10.5 


. I 


25 


5 


9-3 


9-2 


. I 


25 


7 


8.23 


8. 12 


. II 


25 


9 


7-4 


7-34 


.06 


25 


II 


6.82 


6.77 


•05 


30 


5 


12. 2 


12. I 


. I 


30 


7 


II. 7 


11.63 


.07 


30 


9 


11.38 


"•35 


.03 


30 


II 


"■3 


II. 24 


.06 


30 


13 


II. 4 


"-3 


. I 


35 


5 


14.65 


14-53 


. 12 


35 


7 


14.72 


14. 66 


.06 


35 


9 


15 


14-93 


.07 


35 


II 


15-45 


15-32 


•13 


35 


13 


16. I 


15-85 


•25 


40 


8 


17- 57 


17. 62 


- -05 


40 


10 


18.5 


18. 52 


— . 02 



1 62 Journal of Agricultural Research voi. xii.no. 4 

From a study of Tables II, IV, and VI it is evident that the share can 
not be as accurately described by mathematical equations as can the 
moldboard. However, the differences even upon the share are not very 
great. It must be remembered that these surfaces have been developed 
empirically; experience and an extensive knowledge of the conditions to 
be met have been the chief guides. Yet this implement produced in the 
school of experience has a surface approximately mathematically exact 
in form. Further, the surfaces of cast bottoms, which, because of the 
difficulty of manufacture, are not changed unless necessity demands, 
consist in some cases approximately of a portion from each of two quadric 
surfaces. It will be shown later in discussing the history of the plow that 
the surfaces of the Holbrook bottoms were designed to be portions of 
hyperboloids of one sheet. In the Utica (N. Y.) plow trials these 
machines received many first awards and much commendation from the 
judges for the excellence of their work. In addition to this, Mr. J. J. 
Washburn, of Batavia, N. Y., who knew Mr. Holbrook and was 
present at the Utica plow trials, stated that the Holbrook plows did as 
good work as any that it has ever been his pleasure to witness. Thus, 
there is considerable evidence, based upon field experience, which indi- 
cates that a portion of a hyperboloid of one sheet is the proper form for 
the surface of a plow bottom. So far as is known, this hypothesis awaits 
definite proof. 

MOTION OF THE vSOIL PARTlCLEvS IN PLOWING 

For the purpose of studying the motion of the soil particles in plowing, 
the work was limited to sod ground available in the vicinity of Ithaca, 
N. Y. From observations on a sod plow at work in the field (PI. 7, C), 
the following general facts regarding the furrow slice were noted : 

The lower outside * edge of the furrow slice did not appear to be either 
stretched or compressed. 

The upper outside edge of the furrow slice appeared to be compressed . 

The inside of the furrow slice was stretched, the lower edge more than 
the upper edge. 

As the furrow slice passed over the moldboard the cracks, which had 
formed on the inside in traveling over the share and the front portion 
of the moldboard, closed up as the soil passed over the rear of the plow 
bottom, indicating a point of maximum stretching. 

The above considerations made it evident that a more detailed study 
of the behavior of the furrow slice was desirable. For this purpose rows 
of pins were set in the unplowed ground, the pins being driven in the 
ground to the estimated depth of plowing, as shown in Plate 4, A. The 
longitudinal rows are parallel to the line of motion of the plow, which is 
also parallel to the Z-axis (fig. 8) and the transverse rows perpendicular 

'The portion of the furrow slice immediately adjacent to the furrow is called the "outside." 



Jan. 28. 1918 Study of Plow Bottoms 1 63 



to this same line of motion. The longitudinal rows are numbered from 
II to VI (Row I was omitted because the colter upset the pins), and the 
pins in each row numbered from i to 10, as shown in figure 11. When 
the part of the furrow slice in which the pins were set was upon the mold- 
board, it took the form shown in Plate 8, B. In order to obtain the 
X, y, and z coordinates of points in the furrow slice upon the moldboard, 
the apparatus shown in Plate 9, A, was used. In this apparatus the 
axes have the same relation to the plow bottom as those shown in figure 
8. This more detailed study of the furrow slice upon the moldboard 
revealed the following : 

The length of Row II, pins i to 10, on top of the furrow slice was 
greater than the length before the soil had passed upon the moldboard, 
indicating that this por- 
tion of the furrow slice y] 
had been stretched. 

The length of Row 
II, pins I to 10, was 
greater upon the bot- 
tom of the furrow slice 
than its length before 
the soil passed upon 
the moldboard. 

The length of Row 
VI, pins I to 10, on top 
of the furrow slice was P,q ^^ 

less than its length be- 
fore the soil passed upon the moldboard, indicating that this portion 
of the furrow slice had been compressed. 

The length of Row VI, pins i to 10, on the bottom of the furrow slice 

was greater than its length before the soil passed upon the moldboard. 

The lengths of Rows IV and V, pins i to 10, on top of the furrow slice 

w^ere approximately the same as their lengths before the soil passed upon 

the plow bottom, indicating neither compression nor stretching. 

The lengths of Rows IV and V, pins i to 10, on the bottom of the 
furrow slice was greater than their lengths before the soil had passed 
upon the plow bottom. 

The z distances of pin 10 on top of the furrow slice were approximately 
the same for each row, but less than the distance which the plow had 
moved forward. 

The z distances of pin 10 on the bottom of the furrow slice were 
approximately the same for each row and equal to the distance which 
the plow had moved forward. (The coordinates of the pins at the 
bottom of the furrow slice were measured by cutting away a portion 
of the soil but leaving the pins in place.) 




164 



Journal of Agricultural Research 



Vol. XII, No. 4 




These observations reveal, first, that when a cross section of the furrow 
slice is considered (fig. 12) the portion marked "A" is compressed in 
plowing and the portion marked "B" is stretched, while the soil in the 
position of fine /; is neither compressed nor stretched; and, second, that 
there is a definite relation between the z coordinate of a soil particle and 

the distance the plow has moved for- 
ward. This relation is developed on 
pages 164 to 167. 

The next step was to analyze in detail 
the motion of the soil particles. This 
study was limited to the soil particles 
upon the bottom of the furrow slice, but 
the methods developed are applicable to 
other portions. The paths of the soil particles upon the bottom of the 
furrow slice can be very accurately traced from the scratches which 
they make upon the moldboard. Plate 9, B, shows the paths of five soil 
particles. By taking the axes as shown in figure 8, a projection of these 
paths upon the plane z=0 showed a very uniform set of curves. Each 
of these curves (fig. 13) can be very accurately described by equations 
of the general form 

ax^ + hy'^ + lx + my + d = 0. (45) 

When these same paths are projected upon the plane ^- = 0, a set of curves 
resulted (fig. 14), each of which could be very accurately described by 
equations having the following general form : 



Fig. 12. 



ax^ + hz^ + Ixz + m.x + nz + d = 0, 
From equation (45) 



dy , d-y , 

-jr and ,-,- the veloc- 

dt dr 

ity and acceleration, 
respectively, of a soil 
particle in the y direc- 
tion can be found if 

dx d^x 

-jr and --r^ are known. 

di dp 



(46) 



The values of 



dx 
dt 



and 



dH 
df 



can be found from 
dz 




Fig. ij.— Projection of the paths shown in Plate 9, A, upon plane 
z=0. 



equation (46) if -7- and 

d^z 

-T^ are known. Thus, 

to analyse the velocity and acceleration of any soil particle whose path 
upon the surface of the plow bottom is known, an equation must be 
found between z and time (/). 



Jan, 28, 1918 



Study of Plow Bottoms 



165 



This was accomplished by comparing the 2 coordinates of the bottom 
ends of the pins with the distance which the plow had moved forward. 
The distance which the plow moved forward is designated by s, so that 

s = vt, (47) 

where r = velocity of the plow, and / = time. 

By the use of the apparatus illustrated in Plate 9, A, the data given in 
Table VII were obtained for the soil particles upon the bottom of the 
furrow slice whose paths are shown in Plate 9, B. These data are typical 
of 12 sets of observations. 

Table VII. — Values (in inches) of points in thcfiirww slice 



Row II. 


Row III. 


Row IV. 


Row V. 


__ 


i 


z—s 


- 


s 


z-s 


_ 


s 


z—s 




^ 


z—s 


16A 
20 J 

24 
27! 

32* 

35I 
39I 


i5f 

27f 

35| 

39f 


1 

i 

J. 
4 



-i 

-1 


16 

2 of 
23I 
27I 
32i 

35I 

39^ 


i9| 
23i 

27i 

35| 
39f 


1 

4 
5 

"5 



X5l 
195 

24i 

27^ 

35I 

39t 


23| 

3if 
39f 


\ 

-\ 
\ 


i5f 
20I 

23| 

3i| 
35J 
39i 


15I 
i9i 

23f 
27f 

3ii 
35f 
39f 





-h 
-\ 



Unfortunately the soil available in the vicinity of Ithaca was not well 
adapted for taking observations of the kind reported in Table VII. This 
soil is not uniform in texture, contains many stones, cracks much more 
readily than it stretches, and the surface is not as level as could be desired 
for this work. At times it was difficult to drive the pins straight into 
the ground. The data of Table VII show, however, a distinct tendency 
for the difference between z and s to reach a maximum value and then 
decrease again to zero; and also a slight tendency for this maximum 
difference to decrease from Row I to Row V. When the work was begun, 
it was hoped that sufficiently accurate data could be obtained from which 
a law between z and s could be developed, but on account of the difficul- 
ties already explained this was impossible. Consequently, in order to 
develop a method for future work, a set of conditions were assumed which 
agreed qualitatively with the observed facts. It should always be kept 
in mind that this was done simply as an hypothesis whose exactness should 
be thoroughly tested upon a soil better adapted to this work. The 
conditions assumed for the relations between 2 and s are as follows: 

(A) That, for each path, when 2 = 40, 5 = 40. 

(B) That there was no stretching or compression in the outside bottom 
edge of the furrow slice up to the point 2=40. 

(C) That the maximum difference, z-s, for Path I was 1.05 inches.. 



1 66 



Journal of Agricultural Research 



Vol. XII, No. 4 





— < 


V 




i i i 




! 1 












'^ 


^ 


i 1 i 


















\ 


^ 


\ 


i i 




















\\ 
























\ 


w 


^. 
























\\ 


























\ 




N> 


1 






















^ 




>> 


\ 


























\N 


\ 


\ 






















\\ 


V 


A 






















\ 


\\ 


\ 


\ 
























\ 


\^ 


^ N 


\ 


















\ 


\ 


\ 


\ 


\ 


















1 


\^ 


V 


\ ^ 


\ 

\ V 


















\ 


\ 


\ 


\^ 


\ 




















\ 


\ 


\ 


\ 


















\ 


\ 


1 


\ \ 


\ 






















\ 


\ ^ 


\ \ 




















1 




\ 


\ 


\ \ 






















\ 


\ 


\ \ 
















1 i 




\ 


\ 


\ 


\ 






















\ 


\ 


\ 


\ 






















\ 


\ 


























\ 


\ 












i 












\ 




















1 














1 






















































, 












































































1 


1 
























1 




















i 


i 






J 








































i 




































~! 














1 







































































40 (D) That the maxi- 
mum difference, z — s, 

^3 for each path decreased 
uniformly across the 

36 furrow slice. Thus, for 
Row I, re = 0.85 inch, the 

34 maximum 2; — i- = i .05 
inches, and when 
ic=i3.6 inches, the 
width of the furrow- 
slice, the maximum 
z—s=0\ so when 
x=7.5 inches, the max- 
imum z—s for Row V 
is 0.45 inch. 

(E) That the stretch- 
ing in each row took 
place uniformly up to 
the maximum point 
and then decreased 
uniformly until it was 

^'^ zero when 2 = 5 = 40. 

(F) That the maxi- 
'^ mum stretching 

occurred midway be- 
16 tween the point where 

the soil particle passed 
14 upon the plow bottom 

and the point ^• = 40. 
12 Thus, for Path I where 

the soil particle passed 
10 upon the moldboard at 

the point 5- = 0.6: 



32 



30 



28 



26 



24 



22 



8 



Fig. 14. — Projection of the paths shown in Plate 9, A, upon 
the plane y—O. 



40 — 0.6 = 39.4 inches. 
39.4 -^ 2 = 19.7 inches. 
19.7 + 0.6=20.3 inches. 

For Path I the point 
of maximum stretch- 
ing was at .y=2o.3 
inches. 

The computations 
below show that for 
Path V, v/here the soil 



jaa 28.1918 Study of Plow Bottoms 167 

particle passed upon the share at the point s=ii.6, the point of 
maximum stretching occurs at ^=25.8 inches. 

40 — 11.6 = 28.4 inches. 
28.4H-2 = 14.2 inches. 
14.2+ 11.6=25.8 inches. 

The following is the simplest form of a function which meets the 
requirements imposed by the above conditions and, when the constants 
are determined, will describe the relations between z and i- for a soil 
particle on the bottom of the furrow slice as it passes over the surface of 

the plow bottom : 

z-s=^a{s^- + bs+cy (48) 

From equations (47) and (4S) 

z-vt = a[{vty + bvi+cy; (49) 

From (49) -^ and -~, the velocity and acceleration, respectively, of a 

soil particle in the z direction can be obtained. 
From equation (46) by differentiation we have 

{2ax-\-lz+m)£^+i2bz+lx+n)j^ = 0; (50) 

and 

.<^'X , / dx ,dz\dx 
(2ax+/.+ m)^^+(^2a^ + /^j^ 

■d-z / .dz , Jx\dz ^ - . 

+ i,bz+lx+n)^, + l^2b^^ + l^)^^=^0. (51) 

Similarly from equation (45) we find 



and 



,^dx , , , ^dv ^ / V 

{2ax + l)-r-+(2by + m)-fj = 0; (52) 



(2a.. + /)g+ 2a(^;)+ i2by + m/^+ 2.(1)= O. (53) 



^ , , . . dx dy , , 

From equations (50), (51), (52), and (53) the velocities j^, ^-, and the 

accelerations -rv, ,v of a soil particle on the bottom of the furrcw slice 
at at-' '■ 

can be obtained when -jj and j^ ^^^ known. 

In this problem, however, we are interested in the accelerations in the 
directions of the normal to the surface, designated by ".V," the tangent 
to the soil path "T," and the perpendicular to the plane formed by the 
normal and the tangent " R." 



1 68 Journal of Agricultural Research voi. xii, no. 4 

We can find Xj, ^Uj, v^, the direction cosines of the angles which A^ makes 
with the X-, Y-, and Z-axis in either of the following ways : 

If (20) (the equation of the surface of the plow bottom) is known, we 
have by differentiation 

\ ^1 



ax^ + byo + gzQ + l hx^ + byg + fzo + m 



V, 



I 



(54) 



/ 



l{aXf^ + hy^ + gzQ + iy+ {hx^ + by^ + fZf, + mf' 

or if the paths of the soil particles are known but the equation of the 
surface is unknown the angle Ny can be measured by means of a pro- 
tractor and plumb bob, as shown in Plate 9, C. The direction cosines 
Xi and Vy can then be computed from the following: 

(Xir+(Mi)^+K)^=i (55) 

dx dy ^ dz 

^'dt^^Ht-^'^dr^'' ^56) 

where the values for tt, ,' , and ~ can be obtained from (49), (50), and 

(52). 

dx dy d'' 

The direction cosines of T (X,, 1J.2, u,) are proportional to -77, A, and ,"• 

Hence 

X2 At2 ^2 I 



dx dy dz (dx\.(dj\(dz\ (57) 

dt df dt y\dt)'^\dt )^\dt) 

The direction cosines of T (X3, /Xg, v^ can be computed from the follow- 
ing:* 

(X3)'+(M3)'+(^'3)*=I- (58) 

—^ = ^^ - ^^ =±i.' (59) 

^3*^2— i'3M2 Mai'i-'^sA'i Mii^2-i^iM2 

The components in the directions A'^, T, and R of the forces acting on a 
soil element of mass M, moving with the component accelerations 
d}x d^y . d^z 
dP'd^^'^'^df^^'^ 

d'X d^v d^z 
FN = M(X,-i^,+M,^' + i',^) (60) 

FT=M(4f + 4>'+„,|f) (6x) 

Fr=.M(X,^ + 4? + v,^^). (6.) 

■Snydbr, Virgil, and Sisam, C. H. Op. cit., p. 40. 



Jan. 28, 191S 



Study of Plow Bottoms 169 



EVALUATING THE CONSTANTS IN EQUATIONS (48), (46), AND (45) 

The methods of evaluating the constants in equations (48), (46), and 
(45) for a given soil path will now be considered. For this purpose 
Path V (PI. 9, A) will be taken. The general form of equation (48) is 

2-.<r = a(r + 6i + c)^ (48) 

From the assumptions that have already been made (p. 164 to 168) the 
following data for this curve are obtained : 

s 2 

II. 6 1 1.6 

25.8 26.25 

40.0 40-" 

On substituting the above values for .^ and 2 in equation (48), three 

equations are obtained from which it is found that 

a = o.ooooiii4 
6=-5i.6 
£ = 464 

^'"''"^ 2-^ = 0.00001 1 i4(.v2- 51 .65+ 464)' (63) 
To determine the values of the constants in 

ax- + bz' + lxz + mx +n2 + d=0, (46) 

the origin is moved to % = 7.65, ^="-6- For this point as origin an 
equation of the following form describes the curve : 

a{x'y + b{2'y + l,x'z' + m,x' = 0. (64) 

Taking a- I, only three constants, b, l„ and m„ remain to be evaluated. 
From the trace of Path V on the surface of the plow bottom the following 

data were obtained: 

x' 2 

I 13-55 

3 20.05 

6 27.15 

Substituting these values for x' and z' in equation (64) gives three equa- 
tions from which 

b= —0.019 

/i = - 0.453 
Wi= 8.63 
(x')2_o.oi9(zT-o.453x'z' + 8.63x'=a (65) 

Translating the axes back to the original origin, 

x' = x- 7.65 

z' = z— II. 6 

^'^^"^ x^- 0.0192^-0. 453^2- I -45*+ 3- 912-49. 92 = 0. (66) 



lyo Journal of Agricultural Research voi. xii, no. 4 

To determine the values of the constants in 

ax^ + hyf- + lx+my + d=-0, (45 ) 

the origin is moved to, x = 7.6^, >' = o. 2. This changes the form of the 
equation to 

a{xy + b{yy + l,x' + 7ny^0. (67) 

Taking a=i, three constants remain to be evaluated. From the trace 
of Path V upon the surface of the plow bottom, 

x' y' 

I 3-1 

4 5-45 

7 6.68 

Substituting these values of %' and y' in equation (67) gives 

a= I 
6= 4. 29 
/,= -3o.85 
mi= — 3. 67 

(;c0== + 4.29(3^'r-3o.S5x'-3.67/ = O. (68) 

The axes are translated back to the original origin by substituting 

x^x'-j.Ci^ 
y = y'—o. 2 

in equation (68), which gives 

^2H-4.29>'--46. i5x-5.39>/+295.45 = 0. (69) 

Numerical Example 
The surface of a plow bottom is represented by the equation 

0.54a;"— i.-,2y-+ i.i2z- — 3.6gyz— i.62xz + 2.o4xy 
+ 53.6^x+ ii^.goy — 46.4Z+ J^9.4. = 0. 

The motion of a soil particle which passes upon this bottom at the point 
x=().(), y = o.2, 2=9.5 is described by the following equations: 

z = 0.0000 1 622 (5^^ — 45.5^ + 342)^ + s (70) 

— 0.1 192-— 1.126x2+ 20.78%+ 10.032— 201.63 = (71) 

x^+ i.8>'^ — 42.41X— 1. 5_y+ 245.25 = (72) 

s = vt. (47) 



Jan. sS, 1918 



Study of Plow Bottoms 



171 



From equations (70), (71), (72), and (47) the following are obtained: 
Table VIII. — Values (in inches) for — 



s 


z 


X 


y 


18 
27 

36 


iS. 4 

27.4 
36.0 


7-55 
19-5 


3-6 
8.25 
II. 



2=0.00001622(1;-/- — 45. 5'^/ +342 )^+7;f 
^=o.oooo3244[{V-Y--45.5T;/+342)(2i'-/-45.5'z;)j+c; 

^=o.oooo3244[(T;Y--45-S'^^+342)(2i;-)4-(2^'-f-45-5'"^Fl 



dx 
dt'' 



(. 2382+1. 126X-10.03) 



dH 
df^' 



2X — 1. 1262 + 20. 78 

2a;— 1. 1262+20. 78 

dt 3. 63'-!. 5 

dt- 3. 6}' -1. 5 

The plow moved forward with a velocity of 36 inches per second, 

, .^=36/ 

From equations (74), (75), (76), {77), (78), (79), and (80) the 
listed in Table IX are computed. 

Table IX. — Values for — 



(73) 
(74) 

(75) 
(76) 

(77) 

(78) 

(79) 

giving 

(80) 

values 







dz 


dH 


dv 


d^-'v 


dz 


dh 


s 


/ 


di 


dfi 


dt 


dt" 


dt 


df- 




.Sec. 














iS 


H 


7.09 


53-6 


16. g 


28. 4 


37-7 


- 9-07 


27 


H 


25- 15 


47-75 


17-32 


-50.0 


34-44 


— 10. 21 


36 


I 


38-4 


41. 6 


3-44 


-74-8 


36 


29.52 



By making the proper substitutions from (80), (74), (76), and (78) in 
equations (54), (57), (58), and (59) the values of the direction cosines 
for the normals N the tangents to the path T, and the perpendiculars 
to the planes formed by the normals and tangents R for three points 
are computed and listed in Table X. 



172 



Journal of Agricultural Research 



Vol. XII, No. 4 



Table X. — Values of the direction cosines for normals, tangents to the path, and per- 
pendiculars to the planes 

X=7-55 y=3(> 2=18.4 



COS 7Vx= 

cos I^j= 
COS /v j= - 


0-549 
.716 

- -429 


cos T35= 0. 169 
cosTy= .4025 
cos T,= . 9 


cos 2?x= 

cos Ry= 

cos R^= 


0.817 

- • 564 
.0977 


X=ll.5 J^8.2S 2=27-4 


COS N^= 

cos A^y = 

COS Nj= - 


0. 728 

. 229 

- .646 


cos T'x=o. 546 
cos Ty= . 376 
cos T^= . 749 


cos R^= 

cos Ry= 
cos /?j= 


0. 4145 
- -897 
• 149 


1=19.5 >■=" 2=36 


COS iVx= 
COS A^y= - 
COS N^=- 


0.698 

- -215 

- -683 


cos T^=o. 728 
cos Ty= . 065 
cos Te= . 683 


cos R^= 

cos Ry = 

cos /?j= 


0. 102 

- -975 
. 2 



For the purpose of computing the forces a block of soil 2 inches wide, 
1 inch long, and }4 inch thick is taken. The mass of this soil is 



1728.32.2. 12 32.2.12 
p= density. 



(81) 



By the proper substitutions from Tables IX and X into equations (60) , 
(61), and (62) the forces necessary to produce the accelerations are com- 
puted and listed in Table XI. 

Table XI. — Forces necessary to produce acceleration in soil particles 



x= 7-55 


y= 3-6 


2=18.4 


Fn= .00503P 


F'r= .001 1 6p 


Fr= .00252P 


x=ii.s 


>'= 8-25 


2=27.9 


Fn= .002S1P 


Ft= .000248P 


Fa= .00592P 


*=i9-5 


y=ii 


2=36 


Fn= .00234P 


Ft= .oo4c8p 


Fe= .00778P 



A soil particle in passing over the surface of the plow bottom will be 
acted upon by the following : 

(a) A force from the surface of the bottom acting in the direction of 
the normal. 

(b) Gravity. 

(c) Pressure from the weight of the soil above the particle. 

(d) Friction between the particle and the surface. 



Jan. 28. 1918 Study of Plow Bottoms 173 



(e) Shearing, stretching, or compression on each of the remaining five 
sides of the particle, due to its contact with other soil particles. 

The force which produces the movement of a soil particle in any direc- 
tion will be the resultant of the components of the above-listed forces 
which act in the direction of the movement. 

The preceding analysis of the motion which certain soil particles 
have in the operation of plowing has not been developed from as refined 
methods nor as uniform data in all cases as could be desired, but the re- 
sults obtained furnish abundant evidence that the problem here at- 
tempted is by no means hopeless. The study should be continued upon 
a tough sod, which would stretch more uniformly, and some apparatus 
which would remove the necessity of certain soil particles remaining in 
line with each other should be substituted for the pins. 

HISTORY OF THE DEVELOPMENT OF PLOW BOTTOMS 

The Annual Report of the New York State Agricultural Society for 
1867 contains an ex- 
cellent treatise giving 
the geometrical con- 
struction of the sur- 
faces of many histori- 
cal plow bottoms, but 
no attempt has been 
made in that report 

to classify these SUr- ^^^ ,r.^^epcrrUrSro,e.,r,c3oc,.ey 

faces upon the basis ^^^ ^^ 

of their mathematical 

forms. Using the above-mentioned work as a basis, the author has 
attempted to work out the mathematical forms of the most important 
of these historical surfaces with a view to making fundamental compari- 
sons with present-day plow bottoms. 

JEFFERSON'S PLOW BOTTOM 

In 1788 Thomas Jefferson, while making a tour in Germany, devel- 
oped what appears to be one of the first methods recorded for making the 
surface of the moldboard geometrically exact in form.^ He argued that 
the offices of the moldboard were to receive the soil from the share and 
invert it with the least possible resistance. In order to do this, Jeffer- 
son developed a surface which he considered best adapted for the work 
of plowing, but attention should be called to the fact that no evidence 
is offered to prove the assertion. Figure 15 shows the framework for 
generating the Jefferson moldboard, in which lines em and oh are the 
directrices. To generate the surface a straightedge is laid upon eo and 

> GouiD, J. S., et al. Op. cit.. p. 403. 




174 Journal of Agricultural Research voi. xii. No. 4 

moved backward, the straightedge remaining parallel to the plane 
z^O. By taking the point o as the origin, the equation of the surface is 

^by2—2dx2—2bly + 2bdz = 0^ (82) 

b = breadth of furrow 
(i= depth of furrow 
/ = length of moldboard. 

On rotating the XV-axes through tan~^= 2C//36, the equation is 

{gF- + 4(P)y'z- ^bdlx' - 6bHy' + 2bd^gl^+4dFz= 0. (83) 

On rotating the "K'Z-axes through tan"^^^^^, the equation is (96^+ 4(i^) 
[(y"y-{z'y]-Sbdlx' 

+ 2{bd-y/iSb^ + 8d'-3bH-yl2){y" + z'] = 0. (84) 

Translating the axes to the points 

y" = y"+yo 

Z' = Z" +Zo 

where yo has such a value that 

2(9b' + 4d:')yo+2[bd-y/iSb'+8d?-3bH^/^] = 0, (85) 

and Zq has such a value that 

- 2i9b^-\- 4dr-)Zo+ 2[bd^TW+8d^- 3bH^] = 0, (86) 

gives 

{gF^-^4d')[(y''y-(z'y]-8bdlx^+i^-Zo')(9b' + 4d') 

+ (yo+z,)(2bd^i8b' + Sd'-3bH^) = 0. (87) 

Letting the constant terms in (87) equal C gives 

(9b^ + 4d')[(y"y- (2'y-]-8bdlx'+C=0. (88) 

Translating the axes to the point x' = x"-}-Xo where Xq has such a value 
that 

-8bdlxo +C = 
gives 

{gb^-\-4d')[{y"y- {z'y] = 8bdlx". (89) 

This is the equation of a hyperbolic paraboloid.' 

LAMBRUSCHINl'S PLOW BOTTOM 

Lambruschini," an Italian, describes a method for generating the sur- 
face of a plow bottom which he considered to be more efficient than 
the surface developed by the Jefferson method. Lambruschini proposed 

' The method of developing the equation for this surface is given upwn pages 150 to isd- 
* Snyder, Virgil, and Sisam, C. H. Op. cit., p. 73. 
5 IvAMBRUScmNi, R. Op. cit., p. 37-80. 1832. 



Jan. 2S, 1918 



Study of Plow Bottoms 



175 



a helacoid generated as follows: Lay out a rectangle opan (fig. 16) twice 
the desired width of the furrow and of an empirically determined length. 
Take the point m midway between points o and p and draw the line mm 
parallel to pq. A straightedge laid upon mo and moved backward along 
the line mrrii being kept 
parallel to the plane 
z = 0, and with an 
angular rotation pro- 
portional to the move- 
ment toward Wj, gen- 
erates the surface of 
the Lambruschini bot- 
tom. The point of the straightedge which was at a will describe the 
helix oo^q (fig. 16). The equation of this surface is 

y 

— = tan 6, 

X 

where ^ has uniformly increasing values as z increases. 

n 




Fig. 16. 



Then d = f (z), when 0=90 
I = length of line mmi 



radians, 
I 



n_ I 
2 2 



n 



Hence, 



='Kr)- 



(90) 



small's plow bottom ^ 



About 1760, a Scotchman, James Small, established a factory in 

Scotland for the manufacture of plows. The surface of Small's mold- 

^ board is obtained by 

laying a straightedge 
upon op (fig. 17) and 
moving it backward 
parallel to the plane 
2 = 0, with the line pm 
and the curve oh as 
directrices. The equa- 
tion of the curve, a half 
catenary, is obtained by 
drawing a line og (fig. 18) the length of line og (fig. 17). At o erect a line 
00^ perpendicular to Hne og and equal in length to Hne gh (fig. 17). 

' Gould, J. S., et al. Op. cit., p. 415. 




/"rom Report of NXStole Agnc.5oc l$S7 

Fig. 17. 



176 



Journal of Agricultural Research 



Vol. XII. No. 4 



Through point o^ (fig. 18) draw a hne oji parallel and equal to line og. 
With h and as points of suspension describe a catenary with its lowest 
point at O. Taking the point O (fig. 18) as origin, the equation of the 
catenary is ^ 

1/= — (g21z/3ba_j_^— 21z/3ba\ foi) 

a = Og. 
Transferring the origin to the point a gives 



y= — (e-'^ ''^^^ -\- e~2'^ 1"^^^) — a 



(92) 



as the equation of the catenary oh (fig. 17). The equations of line pm 
(fig. 17) are 

y=0 
x=h. 
h 




Fig. j8. 

Any plane parallel to the plane 2 = is given by z^c, and this plane cuts 
the line pm at the point 

y, = 

z^ = c. 

It also cuts the catenary oh at the point 

_3b 

Z2 = C. 

The equation of the line in the plane z = c which cuts the line pm and the 
catenary oh (fig. 17) is 

x—h _ y—0 

(93) 



or 



3* _j l(c)-0 
2/ 

(%-6)/(c)-:^^c-6)=0 



(94) 



Jan. 28, 1918 



Stvdy of Plow Bottoms 



177 



As this line is always parallel to the plane z = 0, it follows that c=^z and 

m=fiz). 

From equations (92) and (94) then, 

(«- &)rf- (e^'^/^*'* + e--'^'^ba) _ ^1 y?^c - 6^ = O, (95) 

which is the equation of Small's moldboard. 

STEPHEN'S PLOW BOTTOM * 

About the same time that Small brought out his moldboard another 
Scotchman named Stephens developed a method for forming the surface 




^p^^ /nj/r? Reporr ofN YStoti Agric 3oc 1867 
Fig. iq. 




From ffeport of NY Slate A<fric^ Soc I6ST 

Fig. 21. 




V -GA 



Fig. 20. 



of a moldboard the general plan of which is shown in figure 19. The 
generator for this surface is a straightedge laid upon op (fig. 19) and moves 
backward parallel to the plane z = with the line on and the curve ph as 
directrices. Stephen designed his surface by taking a quarter cylinder 
opmnhg and laying out p^nii (fig. 20) equal in length to pm (fig. 19). 
Perpendicular to line pitn^ draw m^hi equal to the length of arc 7nh (fig. 
19). Through points p, /^^ (fig. 20) pass a circle of radius 2nb. The plane 
figure p^m^h^h^ (fig. 20) is then laid upon ihe quarter cylinder (fig. 19) 
so that />! falls upon p, my upon m, and h^ upon h. This will locate the 
curve ph (fig. 19), leaving a figure as shown in figure 21. It will be 

»Gotn,D, J. S., et al. Op. cit., p. 431. 



178 



Journal of Agricultural Research 



Vol. XII. No. 4 



observed in figure 2 1 that - = tan 6 where d has gradually increased values 

from O at 2 = to 90° at 2 = /. Further, 6=-r, radians where 7 represents 

the lengths of arcs 11', 22', etc.; then — = tan ( v )• From figure 20 the 
equation of the circle with its center at O, taking p^ as the origin is 



In figure 20 



F=2nb cos <l> 



«^+ 7= 90°; 
B+B'== 90°; 
a + B'+ 7=180°; 
<l) = a-B; 



(96) 
(97) 



G=2nb sin <f> 



'.^P + 



n'b^ 



..b 



(98) 



nb^4n^b^-(^-^+'^) 



J_ 

2nb' 



(99) 



Substituting the values for F from equation (98) and for G from equation 
(99) gives 



(100) 



^ = tan[/(2)]. 

which is the equation of the surface. 

rahm's pi,ow bottom * 

In 1846 Rev. W. L. Rham, an Englishman, brought fonvard the 
theory that the lines of the moldboard running in the longitudinal 

direction should be 
straight, but that the 
section of the mold- 
board formed by any 
plane z = c (fig. 22) 
should be a straight 
line or a cur\^e, ac- 
cording to the phys- 
ical characteristics of the soil to be worked. Mr. Rham agreed that 
for medium, mellow soils the surface of the moldboard should be 




From Repwrt of N. Y. State Agric. Soc. 1867. 
Fig. 22. 



' Gould, J. S., et al. Op. cit., p. 442. 



Jan. 28, 1918 



Study of Plow Bottoms 



179 



generated by laying a straightedge upon oe and moving it backward 
parallel to the plane z = with the lines eji and em as directrices. 
This surface will be a portion of a hyperbolic paraboloid, the same 
general type as the surface which Jefferson proposed. The orthogonal 
projection of the generator in various positions upon the plane ^=0 
will look as shown in figure 23. For stiff, clay soils the lines (fig. 24) 




e o 

From Report of N. Y. State 

Agric. Soc. 1867 

Fig. 23. 




e o 

From Report of N. Y. State 

Agric. Soc. 1867 

Fig. 24 




e o 

From Report of N. Y. State 

Agric. Soc. 1S67 

Fig. 2-. 



are made concave and for loose, sandy soils (fig. 25) they are made 
convex. As no exact description was given regarding the shape of 
the curves (fig. 24, 25), it has not been possible to develop equations 
for the surfaces. However, as it is known that these surfaces have 
straight lines in one direction and can not be described by an equation 
of the second order, they are of the fourth order or higher. 

KNOX's PLOW BOTTOM* 

In 1852 Samuel A. Knox, of Worcester, Mass., applied for a patent upon 
the surface of a plow bottom which was certainly unique. The skeleton 

of this surface is shown 
in figure 26. The seg- 
ments of circles I, 
II, and III are placed 
in parallel planes 12 
inches apart, so that a 
series of straight lines 
will cut the three cir- 
cles. Circles I and III 
have equal diameters 
and the diameter of 
circle II is one-half 
that of circles I and 
III. As the equation of this surface is of the eighth order, it will not be 
worked out in detail, but a development will be given to show how the 
equation could be obtained. 

Let the equation of the three circles be ^ 

^2 + ^2 = iv*2 
z=0, 

z=k: 




from ReponoflitrSlaieAgricSoc /ii67 



Fig. 



1 Gould. J. S., et al. Op. cit., p. 49.';- 

• This development is the work of Virgil Snyder, Professor of Mathematics, Cornell University. 



i8o Journal of Agricultural Research voi. xii. No. 4 

and {%-€)■+ {y-d)- = R- 

Z=2k. 

Draw the line from a point {x^, y\, O) on the first circle to a point {x^, y2, 2k) 
on the third. Its equations are 



x-x^_ y-y\ 



from which 



^2~^i y2~>'i 2k 

2fe(:x: — Xi)+2(Xj — c) 



■ = Xo — c, 



2k{y-y^^rz{y^-d) 
_ __ ^,_^ _ ^ _ 



Since 



{x,-cf^{y,-df = R}, 
we have, after simplifying, 

4^'[(^-^i)'+(y->'i)']+4M(«-^i)(^i-c) 

■^{y-yd{y-d)\^-z\{x,-cf+{y,-df-w\=o. (loi) 

This is the equation of a cone with vertex at {Xy, y^, O) and passing through 
the third circle. 

In the same way, find the equations of the line from (x^, y^ O,) to 
(3C3, ^3, k) on the middle circle 

k(x — x.) + z(x. — a) 
' -x,-a, 

Hy-yi) + Hyi-b) _ _ ^ 

z " 

Since 

we have, after simplifying, 

k\{x-x,f+ {,y-y,Y\^ 2kz\_{x-x,){x,-a) 

^{.y-h){y-y,)\^z^x-af^{y,~hf-~^ = 0. (102) 

When equations (loi) and (102) are multiplied out, it will be seen that 
x^i, y\ always enter in the form 7? ^-\- -f ^^^ R} . By substituting R? for 
*'i + Fi in each, the equations are of the form 

Axy + By^^C, 
A'x,+ B'y, = C'. 



Jan. j8, i9i8 Stiidy oj PI 07V Bottoms i8i 



Solve these equations for x^, y^ and put their values in 

x\ + f, = R\ 
A = [^kz{x+ c) — 2cz^ — 8a;P], 
B = \\hz\y -^d)- 2dz' - 8>//e2], 

C = [4i?'fe2 _ 4^2 (^2 + y2^ _ ^J^^^ _ ^^^y _ ^J^J^2^ ^ ^2 (^2 ^ ^2) J 

A ' = [2kz(x + a) — 2xk^— 2az'], 
B'=^[2kz{y+b) - 2y¥- 2hz% 

C = [R^k'' + k\x^ + y'')- ^kz{ax + hy- R^) + z\ct' + b' + ^R'-)l 

_B' C-BC ' 
^'~AB'-A'B' 

_ C'A-CA' 
^'' AB'-A'B' 
hence {B'C-BC'y+{C'A-CA'f = R\AB'-A'Bf. (103) 

CYLINDRICAL PLOW BOTTOMS 

In 1854 an American, Joshua Gibbs/ patented a plow bottom the 
surface of which is a portion of a circular cylinder. Taking a point upon 
the axis of the cylinder as the origin, the equation of this surface is 

-2 + -,- 1=0 (104) 

In some foreign countries, notably Germany, the hyperbolic cylinder has 
been suggested as suitable for forming the surface of the moldboard. In 
this connection it is interesting to note that any cylindrical surface can be 
described by an equation of the general form. 

mead's plow bottom ^ 

In 1863 a Mr. Mead, of New Haven, Conn., patented a plow bottom, 
the surface of which conformed exactly to a portion of a frustrinn of a 
cone. The general equation of this surface is 

J2 /1/2 .5,2 

-2 + T2-~2-0 (106) 

a^ P c^ 
holbrook's plow bottom 

The Report of the New York State Agricultural vSociety for 1867 con- 
tains a very complete report of the plow trials held at Utica, N. Y., 
in 1867, at which trials a line of plows designed by F. F. Holbrook, of 
Boston, Mass., showed general superiority to all other makes. The 

1 Gould, J. S., et al. Op. cit., p. 502. ^ gquld, J. S., et al. Op. cit., p. 505. 

• Snyder, Virgil, and Sisam, C. H. Op. cit., p. 83. 



1 82 Journal of Agricultural Research voi. xii. no. 4 

following quotation gives a very good description of the Holbrook sur- 
faces : 

We ' were interested in the most minute details of these plows by Gov. Holbrook 
and the trials at Utica and subsequently at Brattleboro, Vt., showed very clearly the 
influence of the warped surface which is generated by his method upon the texture 
of the soil. Gov. Holbrook is as yet unprotected by a patent on his method, and we 
are therefore most reluctantly compelled to withhold a description of it but we have 
no hesitation in saying that it is the best system for generating the true curve of the 
moldboard which has been brought to our knowledge. This method is applicable to 
the most diversified forms of plows, to long or short, to broad or narrow, to high or 
low, no matter what the fonn may be, this method will impress a family likeness 
upon them all. There will be straight lines in each running from the front to the 
rear and from the sole to the upper parts of the share and moldboard. None of these 
lines will be parallel to each other, nor will any of them be radii from a common cen- 
ter. The angle formed by any two of them will be unlike the angle formed by any 
other two; a change in the angle formed by any transverse lines will produce 
a corresponding change in the vertical lines, and there will always, in every 
form of this plow, be a reciprocal relation between the transverse and vertical ^ lines. 
Plows made upon this plan may appear to the eye to be as widely different as it is 
possible to make them, and yet, on the application of the straightedge and protractor, 
it will be found that they agree precisely in their fundamental character. The 
stirface of the moldboard is always such that the different parts of the fiu-row slice will 
move over it with unequal velocities. 

From the above description it is evident that the surfaces of the Hol- 
brook plows are portions of a hyperboloid of one sheet whose general 
equation is 

— jL.y' ——— 

MISCELLANEOUS PLOW BOTTOMS 

In addition to the surfaces already described there remain at least 
three which show unique characteristics, but data were not available for 
developing the equations. 

In I Si 8 Gideon Davis,^ of Maryland, patented the surface of a plow 
bottom which was obtained by using the segment of a circle as a gen- 
erator and two segmerits of another circle as directrices. Somewhat later, 
1834, James Jacobs,^ another American, brought out a plow bottom the 
surface of which was a combination of two mathematical surfaces, each 
of which had sets of straight lines in two directions. 

In 1839 Samuel Witherow, of Gettysburg, Pa., and David Pierce, of 
Philadelphia, Pa., brought out a plow bottom whose surface was gen- 
erated by the most ingenious use of the arc of a cycloid. A more detailed 
description of this plow can be found in the Report of the New York 
State Agricultural Society for 1867.^ 

• Gould, J. S., et al. Op. dt., p. 586. 

' It should be noted that the lines here called transverse are designated as longitudinal (PI. 2, A), and the 
lines called vertical are designated as transverse. 
' Gould, J. S., et al. Op. cit., p. 432. 

* Idem, p. 486. 
*Idem, p. 491. 



PLATE 6 

A. — A plow bottom with two sets of straight lines. 

B. — A plow bottom, the surface of which is composed of each of two surfaces. 

C. — A plow bottom similar to B, but with the surfaces merging into each other 
farther back on the moldboard. 

D. — A plow bottom, the surface of which does not contain an infinite set of straight 
lines. 



Study of Plow Bottoms 



Plate 6 







Journal of Agricultural Research 



Vci.XII, Nu. 4 



study of Plow Bottoms 



Plate 7 





'"Lj£jii" 









Journal of Agricultural Research 



Vol. XII, No. 4 



PLATE 7 



A —A plow bottom with a convex surface which has two sets of straight lines. 
B.-Instrument for measuring the space coordinates of any point of the plow bottom. 
C._A sod plow showing the furrow slice turned by it. 



PLATE 8 

A. — Rows of wooden pins driven into the sod for estimating the stretch of the fur- 
row slice. 

B. — Furrow slice showing the position of the pins when on the moldboard. 



study of Plow Bottoms 



Plate 8 



rJ 



i 



*<?* 




.m^: 









Journal of Agricultural Research 



Vol. XII, No. 4 



study of Plow Bottoms 



Plate 9 




Journal of Agricultural Research 



Vol. XII, No. 4 



PLATE 9 

A. — Plow showing attachment used to obtain tlie x, y, and 2 coordinates of points in 
the furrow slice. 

B. — Moldboard showing the paths of five soil particles. 

C. — Measurement of the angle Ny by use of a protractor and a plumb bob. 



LIBRARY OF CONGRESS 



002 758 503 6 




